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**Bernhard** Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

Prove that if $\displaystyle \sigma$ is the m-cycle ($\displaystyle a_1$ $\displaystyle a_2$ ... $\displaystyle a_m$) then for all i $\displaystyle \in$ {1,2, ... m} we have $\displaystyle {\sigma}^i$($\displaystyle a_k$) = $\displaystyle a_{k+i}$ where k+i is replaced by its least positive residue mod m. Deduce that |$\displaystyle \sigma$| = m.

Can anyone help with this problem? Would be grateful for help!

I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

Peter